Weak commutativity for pro-p groups

Abstract

We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group $$\mathfrak {X}_p(G)$$ generated by two copies G and $$G^{\psi }$$ of a pro-p group, subject to the defining relators $$[g,g^{\psi }]$$ for all $$g \in G$$ . We show for instance that if G is finitely presented or analytic pro-p, then $$\mathfrak {X}_p(G)$$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction $$\nu (H)$$ . We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the $$(n-1)-n-(n+1)$$ Theorem.